Sampling from a target distribution \(\pi \)  is a common task in Bayesian inference. Typically, in machine learning, the (unnormalized) density of the target distribution can be evaluated to approximately sample from it with Monte Carlo, variational, and particle methods [1]}, [2]}, [3]}, [4]}.
We look at a setup that is more common in scientific machine learning and scientific computing, where a sequence of distributions \((\pi ^{(\ell )})\)  is given that converges weakly to a computationally intractable target \(\pi \)  for increasing level \(\ell \rightarrow \infty \) . Here, intractable means that one cannot numerically evaluate the (unnormalized) density of \(\pi \) .
For example, one finds such a setup in Bayesian inverse problems [5]}, [6]}, [7]}, where the target \(\pi \)  corresponds to a posterior distribution that depends on a forward model through the likelihood.
The forward model is typically a system of partial differential equations (PDEs)
for which only numerical solutions can be computed; increasingly more accurate, more expensive discretizations (e.g., mesh width going to 0) of the forward-model PDEs then give rise to a sequence of distributions \((\pi ^{(\ell )})\)  that converges to \(\pi \) .
